The link you posted is not for the timing chain tensioner - it's a guide rail setup for the final drive chain after the transmission. They wear, but slowly and is not the cause of the noise you are hearing.
I replaced my transmission belt not too long ago at 18K miles and the old one was barely worn at all, it could have continued easily another 10K miles, maybe more.
I recently bought three OEM BMW timing chain tensioners so I would be set for the next 20K or so miles, after which I may no longer have the bike, may decide to convert to the hydraulic tensioner, or just keep swapping out old for new mechanical tensioners. I have the crankshaft locking tool and it's an easy job. Definitely get a factory fresh one, either BMW labeled or get the KYMCO equivalent one that is somewhat cheaper, but do not buy a used mechanical tensioner! I think the BMW dealer price is around $90.